I. Introduction. Maxwell's equations require that the Maxwell 2-forms be closed, and hence locally exact. They are globally exact if we discount the e

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1 Debye Potentials for Maxwell and Dirac Fields from a Generalisation of the Killing-Yano Equation. I. M. Benn Philip Charlton 2 Jonathan Kress 3 Mathematics Department, Newcastle University, NSW 2308, Australia. Abstract. By using conformal Killing-Yano tensors, and their generalisations, we obtain scalar potentials for both the source-free Maxwell and massless Dirac equations. For each of these equations we construct, from conformal Killing-Yano tensors, symmetry operators that map any solution to another. mmimb@cc.newcastle.edu.au 2 jkress@maths.newcastle.edu.au 3 philipc@maths.newcastle.edu.au i

2 I. Introduction. Maxwell's equations require that the Maxwell 2-forms be closed, and hence locally exact. They are globally exact if we discount the existence of magnetic monopoles, and Maxwell's equations can then be written as second-order equations for a potential -form. In the absence of sources the Maxwell 2-forms are also co-closed and hence locally co-exact. For charge-free solutions they are globally co-exact and a potential 3-form can be introduced. A Hertz potential is a 2-form in terms of which the Maxwell 2-form can be expressed so as to be simultaneously exact and co-exact, and hence satisfy the source-free Maxwell equations. In certain cases one can parameterise the Hertz potential in terms of a function satisfying a second-order dierential equation. Thus Maxwell solutions can be expressed in terms of a scalar potential: the Debye potential. (Clearly we have paraphrased what Hertz and Debye actually did. Although we shall not attempt to give an historical account it might be noted that Bromwich and Whittaker also contributed to these ideas. We have given references later.) Some of the important solutions to Maxwell's equations in at space are expressed in terms of solutions to scalar equations. If the Lorenz gauge condition is imposed on the potential -form then Maxwell's equations require that it be harmonic. In a parallel basis this requires that its components be harmonic functions. The plane wave solutions may be obtained in this way. The radiating multipole solutions are adapted to spherical symmetry. Here the electric and magnetic elds are required to satisfy a vector Helmholtz equation. Solutions to this vector equation can be expressed in terms of solutions to a scalar Helmholtz equation. Thus the radiating multipole family can be expressed in terms of scalar potentials. In a curved space-time one cannot immediately generalise these solutions, as the derivatives introduce extra connection terms. Cohen and Kegeles [] seem to have been the rst to apply the Hertz potential formalism to Maxwell's vacuum equations in a curved background. They point out that the Hertz potential must be some geometrically privileged 2-form, and that algebraically-special space-times have such 2-forms, corresponding to the repeated principal null directions. Sections II to V are all preliminaries to the Debye potential formalism. A preliminary reading may begin at section VI, referring to the earlier sections as necessary. Section II introduces some notation, whilst Section III expresses some results on the Petrov classi- cation scheme in a form that will be convenient later. In Section IV we introduce the conformal Killing-Yano equation. This conformal generalisation of Yano's Killing-tensor equation was introduced in [2]. We show how the equation can be written equivalently in terms of exterior operations. This provides an elegant statement of the equation. Moreover, it is natural to have the equation expressed in this way when considering Maxwell's equations which are themselves most naturally expressed in exterior form. Any dierential form may be regarded as a tensor on the spinor space. We show how the conformal Killing-Yano equation for a 2-form is equivalent to an equation for a 2-index Killing spinor.

3 This relationship has not always been made clear: indeed, tensors equivalent to Killing spinors have been called Penrose-Floyd tensors. In Section V we consider equations for self-dual 2-forms whose eigenvectors are shear-free. When the 2-forms have only one real eigenvector then Robinson's theorem [3] says that they are proportional to a closed form. We show that equivalently the 2-forms satisfy a `gauged' conformal Killing-Yano equation. This restatement of Robinson's theorem proves convenient for obtaining Debye potentials. When the 2-form has two real shear-free eigenvectors then we recover the equations obtained by Dietz and Rudiger [4]. We obtain the integrability conditions for these shear-free equations that we will need later. We do this by relating the self-dual 2-forms to spinor elds. This is not only a convenient way of obtaining these integrability conditions, but also enables us to consider the Debye scheme for the Dirac equation in section VII. In section VI we show how Debye potentials for Maxwell elds are related to repeated principal null directions in algebraically-special space-times. This had previously been done in Newman-Penrose formalism by Cohen and Kegeles [, 5]. Thus their Debye potential equations were written out explicitly in an adapted basis. The advantage of our approach is that we obtain the equations in a basis-independent way, which we feel makes it easier to see how the various ingredients to the scheme enter. (Of course, to solve the equations in any given space-time necessitates choosing some basis adapted to the geometry.) Other workers have utilised dierent aspects of special properties of space-times admitting Debye potentials. Stewart [6] considered Petrov type D vacuum space-times and used the more specialised Geroch-Held-Penrose formalism to obtain simplied equations. Torres del Castillo [7] emphasises the special properties of totally null foliations or `null strings' in his treatment of null Hertz potentials. Wald [8] has pointed out the nature of the relationship between Cohen and Kegeles' Debye potential equations the decoupled massless eld components of Teukolsky [9, 0] from which he derives the Debye potential equations. In the special case in which there exists a conformal Killing-Yano tensor the Debye potential scheme becomes both simpler and more powerful. (Here our way of writing the conformal Killing-Yano equation is particularly well adapted to the scheme.) It is possible to use Debye potentials to obtain a symmetry operator for Maxwell's equations, mapping any source-free Maxwell solution to another. This has been shown by Torres del Castillo [] using 2-index Killing spinors. Now Kalnins, McLenaghan and Williams [2] have obtained the most general second order symmetry operator for the source-free Maxwell system. Their operator contains a term constructed from a 4-index Killing spinor. We show how such an operator is obtained from the Debye potential scheme. We showhow it is expressed in terms of a `generalised conformal Killing-Yano tensor'. In section VII we treat the massless Dirac system analogously to the Maxwell system. We show the relation to previous work, and in particular show how the Debye potential method generalises Penrose's `spin raising' and `spin lowering' operators, constructed from twistors in conformally-at space [3], to an algebraically-special space-time. 2

4 In the nal section we summarise our results and discuss possible generalisations. II. Notation and Conventions. The exterior calculus of dierential forms will be used extensively. As usual ^ denotes the exterior product and d the exterior derivative. If X is a vector eld then X denotes the interior derivative that contracts a p-form with X to produce a(p )-form. The interior derivative is an anti-derivation on dierential forms (as is d) with X A = A(X) for A a -form. It follows that if! is a p-form then e a ^ X a! = p! where the coframe fe a g is dual to the frame fx a g. On a pseudo-riemannian manifold the metric tensor g establishes a natural isomorphism between vector elds and dierential -forms. The -form X [ is dened such that X [ (Y )= g(x; Y ) for all vector elds Y. Thus X [ has components obtained by lowering those of X with the metric tensor. The inverse of [ is ], resulting in the vector! ] having components obtained by raising those of the -form!. The metric tensor also gives a natural isomorphism between the space of p-forms and the (n p)-forms, where n is the dimension of the manifold. This isomorphism is the Hodge duality map. On a decomposable p-form we have (X [ ^ X 2[ ^ :::^X p [ )=X p X p :::X () where is the orienting volume n-form. We shall only consider four-dimensional Lorentzian manifolds. The metric tensor g will be taken to be positive-denite on space-like vectors. With these conventions we have!=n! if! is of odd degree;! if! is even. The Hodge dual may be combined with the exterior derivative to form the co-derivative d that acts on a form to lower the degree by one. For the case of four dimensions and Lorentzian signature d = d : (3) The exterior derivative and the co-derivative can be expressed in terms of the Lorentzian connection r by d = e a ^r Xa and d = X a r Xa : 3 (2)

5 The Cliord algebra of each cotangent space is generated by the basis -forms. It may be identied with the vector space of exterior forms with Cliord multiplication related to the exterior and interior derivatives by A = A ^ + A ] for A a -form and an arbitrary form. We will juxtapose symbols to denote their Cliord product. Whereas the Cliord product of two homogeneous forms will be inhomogeneous, the Cliord commutator of a 2-form with another form will preserve the degree of that form. The Cliord commutator, denoted [ ; ], is related to the exterior product by [F; ] = 2X a F^X a where F is a 2-form and is an arbitrary form. The Lie algebra formed by the 2-forms under Cliord commutation is the Lie algebra of the Lorentz group. (In general they form the Lie algebra of the appropriate pseudo-orthogonal group.) Cliord multiplication by the volume form relates a form to its Hodge dual. In forming Cliord products it is convenient to denote the volume form by z, that is z = : (4) We then have = z where is the Cliord algebra involution that reverses the order of products and leaves the -form generators xed. Since Hodge duality preserves the space of 2-forms, squaring to minus the identity, the space of complex 2-forms can be decomposed into self-dual and anti-self-dual subspaces satisfying F = if and F = if respectively. It then follows that these sub-spaces form simple ideals in the Cliord commutator Lie algebra. Forms of even degree form a sub-algebra of the Cliord algebra. The complexied even sub-algebra has two inequivalent irreducible representations, the spinor representations. Again we will simply juxtapose symbols to denote the Cliord action on a spinor. So if is a spinor and! any element of the Cliord algebra we write! to denote the Cliord action of! on. The eigenvalues of the volume form label the inequivalent spinor representations. A spinor is even or odd according to whether iz = or iz =. We have a spin-invariant symplectic product, which we will denote by a bracket ( ; ), that is block diagonal on the inequivalent spinor spaces. This product will be chosen such that (u;!v) =(! u; v) : (5) This spinor product gives an isomorphism with the space of dual spinors. We let u denote the dual spinor such that u(v) =(u; v) : 4

6 Since tensor products of spinors and their duals are linear transformations on the space of spinors, (u v)w =(v; w)u; we may naturally identify such tensors with elements of the Cliord algebra. Under Cliord multiplication by an arbitrary form we have Under the involution we have (u v) =u v (u v) = u v: (u v) = v u: (6) The parity of the spinors determines that of their tensor product. For example, if u and v lie in the same spinor space then u v is an even form, as the spinor product is zero on spinors of dierent parity. Equation (6) shows that the symmetry properties of the tensor product determine the eigenvalue of. For example, the symmetric combination u v + v u is necessarily odd under. So if both spinors lie in the same spinor space this combination is then a 2-form. Moreover, this 2-form will be self-dual if both u and v are even. We may expand a tensor product of spinors into p-form components as u v = 4 (v; u)+ 4 (v; e au)e a 8 (v; e abu)e ab + 4 (v; e azu)e a z Here we use the abbreviated notation e ab to denote e a ^ e b. (v; zu)z: (7) 4 Since, in the Lorentzian case, the irreducible representations of the complexied Cliord algebra are the complexications of those of the real Cliord algebra, we may choose the conjugate linear charge conjugation operator such that it is related to complex conjugation by and (u; v) =(u c ;v c ) (u c ) c = u: In general we follow the conventions of [4]. III. Algebraically-Special Space-times. In this section we summarise the Petrov classication of the curvature tensor. In particular, we will state the condition that a space-time be algebraically-special in a form convenient to us. 5

7 In a four-dimensional Lorentzian space-time the Hodge dual map squares to minus one when acting on 2-forms. With the Hodge dual as complex structure the space of 2-forms may be regarded as a three-dimensional complex space. The Lorentzian metric induces a metric on the space of 2-forms. By using the complex structure of Hodge duality this metric denes a complex Euclidean structure on the space of 2-forms. The curvature tensor can be thought of as a map on the space of 2-forms in such a way that, in an Einstein space, it commutes with Hodge duality and may thus be regarded as a complex linear map. It is also self-adjoint with respect to the complex Euclidean structure. The Petrov classication scheme classies the Jordan canonical forms of the curvature tensor. Details can be found in [5] and [6]. The metric tensor induces a metric on the space of p-forms. If and are p-forms then their scalar product is dened by ^ = ; which becomes = 2 X a X b X a X b when and are 2-forms. We may use this metric to regard the tensor product of two 2-forms as an endomorphism on the space of 2-forms; ( )(F )=( F): Clearly, symmetric tensor products correspond to self-adjoint operators, and thus more generally so do tensors with `pairwise interchange symmetry'. Those tensors that are double-self-dual correspond to endomorphisms on the space of self-dual (or anti-self-dual) 2-forms. The curvature tensor may be regarded as an endomorphism on 2-forms by using the metric to relate it to a totally covariant tensor, R =2R ab e ba where fr ab g are the curvature 2-forms. The double-self-dual part of the curvature tensor R + is related to the conformal tensor C and the curvature scalar R by R + = C 6 Re ab e ab = C 3 RI where I is the identity map on 2-forms. The conformal tensor C can be expressed in terms of the conformal 2-forms as C =2C ab e ba : 6

8 Thus acting on an arbitrary 2-form we have C =2X a X b C ab : (8) The Petrov type of a space-time is determined by the number of eigenvectors and eigenvalues of the conformal tensor when acting in this way on the space of self-dual 2-forms. An algebraically-general conformal tensor has three linearly-independent eigenvectors with distinct eigenvalues, all other cases being classed as algebraically-special. Any self-dual 2-form that is null is also decomposable, and hence has one independent real null eigenvector K, K = 0. The principal null directions of the conformal tensor correspond to null self-dual 2-forms that satisfy ^ C = 0. (See [5].) This is clearly satised by anynull eigenform of C. Such eigenforms correspond to repeated principal null directions. An algebraically-special space-time may be characterised as one admitting a null eigenform of the conformal tensor. The only space-times that admit two independent null eigenforms are Petrov type D. In this case the two independent null eigenforms have the same eigenvalue. IV. Conformal Killing-Yano Tensors. Killing tensors were introduced as tensors which obey generalisations of Killing's equation, with conformal Killing tensors obeying analogues of the vector conformal Killing equation. One generalisation was to replace the vector with a totally symmetric tensor, whilst Yano [7] extended Killing's equation to a totally anti-symmetric tensor. We shall (as is now common) refer to the latter as Killing-Yano tensors, reserving the term Killing tensor for a totally symmetric tensor. Killing's equation expresses the condition that a vector eld generate an isometry in terms of the symmetrised covariant derivative of the vector eld. The Killing tensor and Killing- Yano tensor equations are also usually expressed in terms of the symmetrised covariant derivative. However, the eciency of the exterior calculus enables the Killing-Yano equation to be written much more succinctly in terms of the anti-symmetrised covariant derivative. If K is any vector eld then the covariant derivative rk [ can be decomposed into symmetric and skew parts. The symmetric part can be further decomposed into a trace-free part and the trace. The symmetrised covariant derivative is related to the Lie derivative of the metric tensor. In four dimensions we have r X K [ = 2 X dk[ 4 X[ d K [ + 2 L K g 4 Tr (L Kg)g (X) where L K denotes the Lie derivative and Tr the trace. (As is usual, we use the metric 7

9 tensor to regard any degree two tensor as a linear map on vector elds.) So the conformal- Killing equation can be written (in four dimensions) as r X K [ = 2 X dk[ 4 X[ d K [ 8X: (9) If in addition d K [ =0then K is a Killing vector. Notice that (9) implies that e a ^r Xa K [ = 2 ea^x a dk [ = dk [ : Since e a ^r Xa =d the coecient of X dk [ is just such that we cannot conclude that dk [ =0. Similarly (9) implies that X a r Xa K [ = 4 X a (e a d K [ )= d K [ in four dimensions. Thus the coecient of X [ d K [ is just such that we cannot conclude from (9) that d K [ = 0. This observation suggests how the conformal Killing equation (9) can be generalised to forms of higher degree: the covariant derivative is related to the exterior derivative and the co-derivative with the coecients chosen such that we do not automatically have the form closed or co-closed. If! is a 2-form then this generalisation gives the equation 3r X! = X d! X [ ^ d! 8X : (0) From this we can use r X!(Y;Z)= 2 Z Y r X!to show that! must also satisfy r Y!(X; Z)+r Z!(X; Y )= 3 d!(x)g(y;z) 6 d!(y )g(z; X) 6 d!(z)g(x; Y ) 8X; Y; Z ;() which is Tachibana's conformal generalisation of Yano's Killing equation [2]. Now taking X = X a and Y = X b in equation (), and multiplying both sides by e ab, we recover equation (0). Hence (0) and () are equivalent and we may adopt (0) as the conformal Killing-Yano (CKY) equation. The CKY equation is invariant under Hodge duality. Equation () shows that (X [ ^ d!)= X d! = X d! by(3) = X d! by (2). Equivalently (X d!) =(X [^d!) : 8

10 Since r X = r X it follows that! is a CKY tensor if! is. Thus any solution to the CKY equation can be decomposed into self-dual and anti-self-dual CKY tensors. The CKY equation also often appears in yet another guise. Elements of the Cliord algebra are naturally identied with tensors on the space of spinors. More generally, tensor products of exterior forms may be regarded as higher degree tensors on the spinor spaces, and so any equation for an exterior form can also be written in spinor notation. The CKY equation can be written as where with =0 = a e a a =3r Xa! X a d! + e a ^ d!: The tensor can be thought of as a tensor acting on three even spinors u; v and w, and one odd spinor by (u; v; w; )=(u; a v)(; e a w) : Since the spin-invariant product satises (5) we have (u; a v)= ( a u; v) for a 2-forms, =(v; a u) since the product is symplectic. Thus is automatically symmetric in the rst two spinors. It will be totally symmetric in the three even spinors if it is symmetric under interchange of u and w say. Since the space of even spinors is two-dimensional and the product is symplectic we have the identity Thus Now (u; v)w +(v; w)u +(w; u)v =0 for all even u; v; w. (u; v; w; )=(u; a v)(; e a w) = ; e a f( a v; w)u +(w; u) a vg =(v; a w)(; e a u) (w; u)(; e a a v) =(v; w; u; ) (w; u)(; e a a v) =(w; v; u; ) (w; u)(; e a a v) : e a a = e a ^ a + X a a ; and the a are such that e a ^ a = 0 and X a a = 0. Thus is totally symmetric on the three even spinors. If! is self-dual then (u; v; w; ) will be zero unless the rst three 9

11 spinors are even and the last is odd. Thus the CKY equation for a self-dual 2-form can be regarded as a Spin-irreducible tensor-spinor equation. This spinor equation, equivalent to the CKY equation, was introduced in its own right in [8], and is now usually known as the Killing spinor equation [3]. Because Killing spinors were introduced separately their correspondence with CKY tensors has not always been made clear. There is potential for confusion in that tensors corresponding to Killing spinors have also been called Penrose- Floyd tensors [9]. In the following section we will consider equations for 2-forms related to shear-free congruences. The CKY equation can be regarded as a special case of the shear-free equation. Integrability conditions for the CKY equation then follow as special cases of those for the shear-free equation which will be derived in the following section. V. Shear-free Equations. A congruence of curves may be specied by a vector eld, the tangent eld. The congruence is shear-free if the tangent eld generates conformal transformations on its conjugate space (the space of vectors to which it is orthogonal). Thus the shear-free condition is a generalisation of the conformal-killing condition. Since the shear-free condition is conformally invariant, the various shear-free equations that will be given all have a conformal covariance. If a vector eld generates conformal transformations on its conjugate then so does any vector eld proportional to it. Since a reparametrisation of the congruence corresponds to a scaling of the tangent eld the shear-free condition is a reparametrisationinvariant property of the congruence. The condition that a congruence be shear-free can be formulated as a `gauged' conformal Killing equation, where the connection terms ensure covariance under a scaling of the vector eld. Thus avector eld K corresponds to a shear-free congruence if it satises the equation [20] Here b r is a scaling-covariant derivative, br X K [ = 2 X ^dk [ 4 X[ ^d K [ 8X: (2) br X K [ = r X K [ +2qA(X)K [ for some real -form A. (The factor of 2 and the constant q will be convenient later.) The gauged exterior derivative ^d and co-derivative ^d are related to b r by ^d = e a ^ b rxa ^d = X a b rxa : (Equation (2) has the numerical coecients chosen for four dimensions, although it is easily generalised to arbitrary dimensions.) Throughout we shall write various shear-free 0

12 equations in terms of a `gauged' covariant derivative. However, it must be remembered that the form A, playing the role of connection, is not some given background eld, but depends upon the vector eld K. When K is non-null then A can be expressed in terms of K, whereas in the null case only certain components can be expressed in terms of K [20]. If K is null (and real) then it may berelated to an even spinor u by K [ =(iu c ;e a u)e a : Clearly the correspondence between K and u is not one-to-one, there being a U() freedom in the choice of u. The shear-free condition (2) for K is then equivalent to the following equation for u: where br X u 4 X[ ^Du =0 8X (3) br X u = r X u + qa(x)u (4) where A is a complex -form whose real part is A, and ^D is the Dirac operator ^D = e a b rxa : The shear-free spinor equation (3) is a C -covariant twistor equation, where C is the group of non-zero complex numbers. The U() part of the covariance stems from the projective relationship between u and K, whilst the scaling part of the covariance is related to the reparametrisation-invariance of the shear-free condition. In the same way that we showed the spinorial correspondence of the CKY equation in the previous section, we may regard (3) as an equation for a spin tensor acting symmetrically on two even spinors and one odd spinor. Written thus the shear-free spinor equation was obtained by Sommers [2]. A (real) null vector eld K may be put into correspondence with a self-dual decomposable 2-form by the relation K [ =0: A given K only determines up to a complex scaling. The shear-free condition for K gives rise to an equation for. In fact Robinson's theorem [3] shows that is proportional to a closed (and hence, since it is self-dual, co-closed) 2-form; that is, a Maxwell solution. It will be convenient for us to state the shear-free condition for dierently. In terms of the even spinor u representing K we may choose = u u: (5) The shear-free condition for u then translates to an equation for. It is simplest to obtain the corresponding equation for written in terms of a Cliord commutator, [ ; ]. If we write as in (5) then we can show that u satises (3) if and only if br X 4 [X[ e a ; b rxa ]=0 (6)

13 where br X = r X +2qA(X): (7) Since the Cliord commutator term can be written as 2 [X[ e a ; b rxa ]=X ^d X [ ^ ^d b rx we see that the shear-free equation (3) is equivalent to the gauged conformal Killing-Yano equation 3b rx = X ^d X [ ^ ^d : (6 0 ) (We reiterate that the form A depends upon the 2-form : in particular, should there be two shear-free 2-forms then, in general, the `gauge terms' occurring in each equation will be dierent.) Thus as an alternative to Robinson's theorem we can state that a decomposable self-dual 2-form corresponds to a shear-free null congruence if and only if it satises the C -gauged conformal Killing-Yano equation. One can show that null solutions to (6 0 ) can be scaled to produce Maxwell elds, and vice versa, and so indeed this statement is equivalent to Robinson's theorem. Previously Dietz and Rudiger [4] investigated a generalisation of Robinson's theorem. They considered non-decomposable 2-forms corresponding to two independent null congruences. They showed that both these congruences are shear-free if and only if the self-dual 2-form satises an equation equivalent to (6 0 ). Although they showed that their equation was a generalisation of the CKY equation they did not interpret the extra terms as gauge terms. Moreover, they only considered non-decomposable solutions to the equation. In fact it is rather nice to have a single equation for a 2-form that characterises any eigenvectors as being shear-free, whether there be one or two independent real eigenvectors. The shear-free equations can be dierentiated to obtain integrability conditions relating second derivatives to curvature terms. We shall make use of these later on. Firstly we consider the shear-free spinor equation. Dierentiating (3) introduces the curvature operator ^R(X; Y ) of b r. Since b r is related to r by (4) the curvature operators are related by ^R(X; Y )u = R(X; Y )u qx Y Fu (8) where F is the C curvature, Since F = da : R(X; Y )u = 2 ea (X)e b (Y )R ab u (9) where R ab are the curvature 2-forms, equation (3) has the integrability condition R ab u +2qX b X a Fu 2 (e b b r Xa e a b rxb ) ^Du =0: (20) 2

14 Multiplying by e a gives P b u 2qX b Fu + b rxb ^Du + 2 e b ^D 2 u =0 (2) where P b are the Ricci -forms [4]. Multiplying this by e b produces where R is the curvature scalar. Hessian, ^D 2 u = 4q 3 Fu Ru (22) 3 A Laplacian on spinors is given by the trace of the ^r 2 = b rx a b rxa b rrx a Xa : This is related to the square of the Dirac operator by curvature terms: ^D 2 u = ^r2 u + 2 eab ^R(Xa ;X b )u: (23) By (8) and (9) we have e ab ^R(Xa ;X b )u= 2 Ru+2qFu and so (22) can be written as ^r 2 u = q 3 Fu 2 Ru: (220 ) Since the conformal 2-forms C ab are given by C ab = R ab 2 (P a ^ e b P b ^ e a )+ 6 Re a^e b we may use (20), (2) and (22) to obtain the integrability condition C ab u + q ab u =0 (24) where ab = 6 e baf + 2 Fe ba : (25) Now we look at the integrability conditions for the decomposable self-dual 2-form describing the shear-free congruence. If is related to u by (5) then we may use the integrability 3

15 condition (24) of (3) to obtain an analogous integrability condition of (6 0 ). If we dene C by (8) then from (5) and (7) C = 2 (u; e abu)c ba = 2 (u; C abu)e ba by the `pairwise symmetry' of the conformal tensor. Now we may use (24) to obtain C = q 2 (u; abu)e ba from (25), = q 4 (u; ( ab ab )u)e ab : = q 2 (u; [F;e ba]u)e ab = q 2 (u; e abu)[f;e ab ] due to the `pairwise anti-symmetry' of X p X q [F;e ba ], from (7). = 2q [F;] (26) 3 In analogy with (23) the trace of the Hessian is related to the `gauged' Laplace-Beltrami operator ^4 by where ^4 = ^r2 + e b ^ X a ^R(Xb ;X a ) = ^r2 + 4 [eba ; ^R(Xb ;X a )] ^4 = ( ^d ^d + ^d ^d): Since b r is related to r by (7) the curvatures are related by and hence ^R(X; Y ) = R(X; Y ) 2qX Y F [e ba ; ^R(Xb ;X a )]=[e ba ;R(X b ;X a )]+4q[F;] 4 = 2C R +4q[F;]: 3 Thus ^4 is related to ^r 2 by ^4 = ^r 2 2 C R + q[f;]: (27) 3 4

16 By dierentiating (6 0 ) we obtain 3 ^r2 = ^4; and so (26) gives the integrability condition or ^r 2 = q 3 [F;] R (28) 6 ^4 + 2 R = q[f;]; (280 ) where the left-hand side is the conformally covariant (gauged) Laplace-Beltrami operator. The integrability condition (26) shows that ^ C = 0 and thus the null corresponds to a principal direction. In a Ricci-at space-time the Goldberg-Sachs theorem makes the stronger statement that a shear-free null congruence must correspond to a repeated principal null direction, and conversely any repeated principal null direction must correspond to a shear-free congruence [22, 23]. More generally necessary and sucient conditions for a shear-free congruence to correspond to a repeated principal null direction are given by the generalised Goldberg-Sachs theorem [24]. So in the generalised Goldberg-Sachs class of space-times must be an eigenform of the conformal tensor. Thus from (26) the commutator of F and must be proportional to, and so from (5) we see that the spinor u must be an eigenvector of F. Thus qfu =3u ; (29) for some eigenfunction, and q[f;]=6 : (30) The integrability condition (22) for the spinor becomes ^D 2 u =4u ^r 2 u = u Ru 3 (3a) Ru; 2 (3b) whilst (28) becomes ^r 2 =2 ^4 + R =6 ; 2 (32b) R (32a) 6 and (26) becomes C =4 : (33) 5

17 We can obtain integrability conditions for the CKY equation using the results that we have for shear-free equations. In the case of a null self-dual CKY tensor we simply have the conditions above where the `gauge' terms are zero. In this case the corresponding shear-free spinor equation reduces to the twistor equation. From (26) we see that the CKY tensor is anull eigenvector of C with eigenvalue zero. In fact (24) shows that any self-dual 2-form that has u as eigenspinor satises C = 0. Therefore C must have two linearly-independent eigenvectors, and so the space-time must be type N or conformally-at. If! is a non-null self-dual 2-form then we may write! in terms of a pair of even spinors u and u 2 as! = 2 (u u 2 + u 2 u ) : (34) In the same way that we showed that the shear-free spinor equation (3) led to the `gauged' conformal Killing-Yano equation (6) we may show that the non-null! satises the CKY equation if and only if the spinors fu i g both satisfy a shear-free equation, with the spinors having opposite C `charges'. (Dietz and Rudiger showed that the 2-form! satises a gauged CKY equation if and only if the spinors fu i g satisfy the shear-free equation [4]. In general the C `charge' of! is the sum of those of u and u 2. So we have here just a special case when! is a CKY tensor.) We may use the integrability conditions for the spinor equations to obtain integrability conditions for!, just as we did in the shear-free case. We have C! = 2 (u 2;e ab u )C ba = 2 (u 2;C ab u )e ba = 2 (u ;C ab u 2 )e ba : The integrability condition (24) for u gives whilst that for u 2 gives C! = q 2 (u 2; abu )e ba ; (35) since q 2 = q. C! = q 2 2 (u ; abu 2 )e ba = q 2 (u 2; ab u )e ba 6

18 Subtracting this from (35) shows that u 2 ; ( ab ab )u e ba =0: Repeating the steps that lead to (26) then shows that we must have [F;!]=0: (36) From this we can conclude that the self-dual part of F is proportional to!. Since u and u 2 are eigenvectors of!, and hence F, there must be some function such that q i Fu i =3u i (no sum.) (29 0 ) Both of the fu i g satisfy the integrability conditions (3a) for the appropriate covariant derivative. From each of the spinors we can make a null shear-free 2-form: i = u i u i (no sum,) each satisfying the integrability condition (33). Thus each of the i corresponds to a repeated principal null direction. Thus for a non-null CKY tensor to exist we must have Petrov type D (or conformally-at), within the generalised Goldberg-Sachs class of spacetimes. Each of and 2 satises the integrability conditions (32a). We now return to the integrability condition (35) for!. If we insert (25) into (35) and use (29 0 ) then we have C! = 8! : (37) Thus, 2 and! form an eigenbasis of self-dual 2-forms under the action of the conformal tensor. The CKY tensor! also satises (32a) where the `gauge terms' are absent from the Laplace-Beltrami operator. We have seen that for a non-null CKY tensor to exist the space-time is necessarily Petrov type D. Suppose now that we have Petrov type D with and 2 null shear-free 2-forms corresponding to repeated principal null directions. We cannot, in general, assume any relation between the gauge terms in the two dierent shear-free equations. We let! be dened by (34) such that f ; 2 ;!gis an eigenbasis for C. Since each i is an eigenvector of C (29) becomes q i F i u i =3 i u i (no sum,) (29 00 ) where now we cannot assume that F and F 2 are related. It then follows that each i is an eigenvector of C with eigenvalue 4 i, and since the eigenvalues are assumed equal we 7

19 can conclude that = 2. Proceeding as we did in (35) we can express C! in terms of the integrability conditions for either spinor: C! = = 4 (u 2;e ba u ) 4 (u 2;e ba u ) 4 (q F u 2 ;e ba u ) 4 (q 2F 2 u ;e ba u 2 ) Since! has been assumed an eigenvector of C we must have u 2 an eigenspinor of F. By taking the product with u, which is also an eigenspinor, we see that the eigenvalues of the two spinors must be opposite. We can draw similar conclusions for F 2 and have e ba e ba : q F u =3u q F u 2 = 3u 2 q 2 F 2 u = 3u q 2 F 2 u 2 =3u 2 : It follows that the self-dual part of q F + q 2 F 2 is zero. That is, the self-dual part of the curvature of the gauge term entering into the equation for! is zero (as was shown by Dietz and Rudiger [4]). It then follows that! satises (27) with the C -curvature term on the right-hand side zero. Hence! satises (32a), and in addition (30). Thus (32a) is satised by any shear-free 2-form whose eigenvectors are aligned with repeated principal directions. We have seen that in type D space-times the self-dual part of the C -curvature associated with! must vanish. When the anti-self-dual part is also zero then! can be scaled to a CKY tensor. Penrose and Walker's result on Killing spinors shows that this can be done in every type D Einstein space [8]. VI. Debye potentials and symmetry operators for source-free Maxwell elds. Maxwell's equations require that the electromagnetic eld 2-form F be closed. This is automatically satised if F is exact, F = da say, where A is the potential -form. In source-free regions F is also required to be co-closed, that is d F = 0. This would be automatically satised if there were a co-potential 3-form B such that F = d B. Suppose that a 2-form Z satises 4Z = dg d W; (38) where G and W are arbitrary forms of degree one and three respectively. Then by noting that this can be rewritten as d (W dz)=d(d Z G) (38 0 ) 8

20 we see that Z provides us with a 2-form that is both exact and co-exact and hence a Maxwell solution. The 2-form Z is called a Hertz potential [25]. On the face of it (38) doesn't oer a promising approach, solving this equation is no easier than solving the original Maxwell equations. However, in some circumstances, with an appropriate choice of the form for Z, solutions to (38) can be written in terms of solutions to a scalar equation; a Debye potential equation [26, and references therein]. Cohen and Kegeles [] applied the Debye potential method to the solution of Maxwell's equations in curved space-times. They pointed out that in algebraically-special space-times there exist privileged 2-forms corresponding to the repeated principal null direction. Using the Newman-Penrose formalism they explicitly obtained scalar Debye potentials by aligning the Hertz potential with the repeated principal null direction: the resulting scalar equation being expressed in the adapted null basis. In this section we show how a Debye potential is obtained by choosing Z to be proportional to a shear-free 2-form in an algebraically-special space-time. As we shall explicitly use the shear-free equation, rather than an adapted basis, the resulting Debye potential equation will be expressed in a basis-independent form. The scheme becomes more powerful when a conformal Killing-Yano tensor exists. Consider a shear-free 2-form, satisfying (6 0 ). (We allow to be either null or non-null, and in the case in which the C -charge is zero then we have a CKY tensor.) Let f be a scalar eld with opposite C -charge so that the 2-form f is a C -scalar. Then the Laplacian of f can be expressed as r 2 (f)= ^r2 f+2^rxa f^rx a+f^r2 where ^r is the C -covariant derivative. We can write this in terms of the the C -gradient as or r 2 (f)= ^r2 f+2^rdgradf + f ^r2 ; 4(f)= ^r2 f+2^rdgradf + f ^r2 by (27). Now, from the shear-free equation (6 0 ) we have 2 fc fr (39) 3 3 ^rdgradf = d gradf ^d ^df ^ ^d = X a ^rxa (f ^d) f ^r Xa ^d d(f ^d )+f^d^d = d (f^d)+f^d ^d d(f ^d )+f^d^d = d (f^d) d(f^d ) f^4: Inserting this into (39) gives 2 4(f)= ^r2 f 3 d f ^d 2 3 d f ^d + f ^r ^4 2 fc 3 fr: 9

21 When is self-dual with its eigenvectors repeated principal null directions this becomes, by (32a), 4(f)= ^r 2 f 6 Rf f 2 3 d f ^d 2 3 d f ^d where C = : (40) It is convenient to identify as an eigenvalue, C = n 2 ; (40 0 ) where n is the number of (real) eigenvectors of. Thus when is null n = and is just the eigenvalue of C corresponding to, whereas in the non-null case n =2and is a quarter of the eigenvalue of C. So if the scalar eld satises the equation ^r 2 f then we can relate an exact form to a co-exact one: d d 2 2 (f) 3 f ^d = d 3 f ^d d(f) 6 Rf = f (4) Thus out of the 2-form and the scalar f we have a source-free Maxwell solution, F 0 (f; ) =d d 2 (f) 3 f ^d = d 2 3 f ^d d(f) : (42a) : (42b) Notice how with this approach, unlike that of Cohen and Kegeles, we do not need to choose the gauge terms G and W by inspection, rather they appear naturally and are given explicitly in term of. Using (3) and the fact that ^d = ^d, we can see that F 0 (f; ) isanti-self-dual for self-dual. To apply this scheme to construct a Maxwell solution in a given space-time it is necessary to solve the scalar equation (4). The shear-free 2-form enters into this equation, not only through the eigenfunction, but also through the `gauge term' A. We may expose the gauge terms in the shear-free equation (6 0 ) by writing it as where we have dened Y X by Y X = q[x [ ^A;] 4qA(X) (43) Y X =3r X X d + X [ ^ d : (44) 20

22 When there is only one repeated principal null direction then the self-dual will be null. To use (43) to determine A it is convenient to pick some adapted basis. (Such a point comes to us all when we have to actually solve equations.) Let be the maximal totally isotropic subspace such that X 2, X [ =0. Then for X in (43) reduces to Y X = 2qA(X) 8X 2 : This enables half of the components of A to be found. To compute the remaining components we may pick a maximal isotropic subspace 0 to complement. Then let be a null self-dual 2-form such that X 2 0 () X [ =0. Then the remaining components of A can be calculated from (43) which becomes Y X = 6qA(X) 8X 2 0 : In a type D space-time there are two repeated principal null directions. Thus one can choose either of the two corresponding null shear-free 2-forms as Hertz potentials, or one can choose the non-null 2-form! that has both repeated principal directions as eigenvectors. Thus there are three possible choices of Hertz potential, as was pointed out by Mustafa and Cohen [27]. They chose to normalise! to have constant length, whereas we have chosen to scale! to be CKY and so their equation for a non-null Debye potential will diers from ours by exact gauge terms. In the non-null case the gauge term is determined directly in terms of!. In this case (43) gives Y X!! = 4qA(X)!! 8X : This expression was given by Dietz and Rudiger [4]. Clearly the scheme simplies in the case in which wehave a CKY tensor, for then the gauge terms are absent from the Debye potential equation. Equation (42a) enables a Maxwell eld to be constructed from a shear-free 2-form and a scalar Debye potential satisfying (4). It turns out that we can conversely take a shear-free 2-form and a Maxwell eld and construct a scalar Debye potential. If f(f; ) =Fthen the C -covariant Laplacian is given by ^r 2 f(f; ) =r 2 F+2r Xa F ^rx a+f ^r2 ; where the C -charge of f(f; ) is the same as that of. Since satises the shear-free equation (6 0 ) 3r Xa F ^rx a = r Xa F (X a ^d) rxa F (e a^ ^d ) =df ^d + d F ^d =0 2

23 by Maxwell's equations. We may then use (27) to relate r 2 F to 4F, which is zero by Maxwell's equations, to obtain ^r 2 f(f; ) = 2 CF + 3 RF + F ^r2 = F 2 C + 3 R + ^r2 since C is self-adjoint, by (32a), = F 2 C + 6 R +2 = F ( + 6 R) : So f(f; ) satises equation (4): notice, however, that f(f; ) has the same `charge' as, whereas the construction of a Maxwell eld from requires a Debye potential with opposite charge. For those space-times that admit a non-null CKY tensor the above gives a method of mapping any Maxwell eld to another. As was shown in section 5, when we have a nonnull CKY tensor! we have a pair of null shear-free 2-forms, and 2, having opposite C -charge. We can take one of these, say, and a given Maxwell eld F to construct a Debye potential f(f; ). This will then have the appropriate charge to combine with 2 to form a new Maxwell eld (also note that = 2 ). That is, we have a symmetry operator L 2, mapping between Maxwell elds, dened by L 2 F = F 0 (F ; 2 ): Interchanging the roles of and 2 gives another symmetry operator. However, by using the shear-free equations for and 2 it can be shown that when acting on a Maxwell eld F, their dierence vanishes and so L 2 F = L 2 F: We can also use a non-null CKY! directly to make a symmetry operator for Maxwell elds. The scalar f(f;!) is a Debye potential satisfying an uncharged equation and hence can be combined again with! to produce a Maxwell eld. Hence L!! F = F 0 (F!;!) is another symmetry operator. Since self-dual and anti-self-dual 2-forms are mutually orthogonal, these symmetry operators map only the self-dual part of a Maxwell eld to an 22

24 anti-self-dual Maxwell eld. To see the relationship between these symmetry operators, we will rst recast them in terms of a higher order generalisation of a CKY 2-form. The CKY tensor! enters quadratically in the symmetry operator L!!. By taking the tensor product of! with itself we obtain a tensor, quadratic in!, which we may regard as an endomorphism on the space of 2-forms, as we did in section 3. In this way we may write L!! in terms of a degree-four tensor constructed from!. Let P + be the operator that projects out the self-dual part of any 2-form. Then clearly P + commutes with the Hodge dual. Since it is self-adjoint with respect to the metric on 2-forms it corresponds to a pairwise-symmetric tensor. Then out of the self-dual! we construct the tensor K: K =!! 3 (!!)P + : (45) In section 4 we showed how! corresponded to a spin-tensor. In the same way one can show that K corresponds to a totally symmetric spin tensor. The symmetry operator constructed from! can be written in terms of K as L!! F = d d 2 (KF) 5 D K(F) (46a) where the `exterior derivative' D is dened by = d 2 5 DK(F) d(kf) DK(G) =d(kg) e a ^ K(r Xa G) for G an arbitrary 2-form. The `co-derivative' D is dened analogously. (46b) The only non-zero inner products between a CKY! and its associated oppositely charged null shear-free 2-forms and 2 are related by 2 = 2!!: After calculating the action of K on the self-dual basis f!; ; 2 g we can use this to see that an alternative expression for K is K = P + : Then using the shear-free equations for and 2 we can rewrite the right hand side of (46a) to show that L!! F = 2 L 2 F + 2 L 2 F = L 2 F = L 2 F; 23

25 since we have already seen that the two terms on the right hand side are equal when acting on Maxwell elds. Hence, as was pointed out by Torres del Castillo [] who wrote down these operators using the two-component spinor formalism, the various Debye schemes give rise only one symmetry operator. The CKY equation for! can be used to obtain an analogous equation for K. The analogy is closest if we write the CKY equation in terms of the Cliord commutator as in (6). For any 2-form let L be the operator that maps any 2-form to the Cliord commutator: L ( ) =[; ]. Then the CKY equation can be written as whilst K satises the equation r X! 4 L X [^e ar X a! =0; r X K 6 [L X [^e a; r X a K]=0: (47) Here the bracket denotes the commutator of the operators. One can show that equation (47) corresponds to a spin-irreducible spin-tensor equation, as we showed is the case for the CKY equation. Just as the CKY equation corresponds to the Killing spinor equation, equation (47) corresponds to the `4-index Killing spinor' equation. Kalnins, McLenaghan and Williams obtained a symmetry operator for Maxwell elds from a 4-index Killing spinor [2]. They then obtained a corresponding tensor equation which they observed was analogous to the CKY equation as written by Tachibana [2] (equation ()). VII. Debye potentials and symmetry operators for massless Dirac elds. In the previous section we showed how we could associate a Debye potential with a shearfree 2-form, enabling the source-free Maxwell equations to be solved in terms of solutions to a scalar equation. In the more special case in which there existed a CKY tensor, we showed the relation between the Debye potential scheme and a symmetry operator constructed from the CKY tensor. In this section we shall show the analogues of these constructions for massless Dirac elds. Let u be an even shear-free spinor corresponding to a repeated principle null direction, and let f be a scalar eld with opposite C -charge. Then the odd spinor 0 (f; u) given by 0 (f; u) = ^dfu + 2 f ^Du (48) is a C -scalar. The action of the Dirac operator on 0 (f; u) is D 0 (f; u) =e a brxa^dfu + ^df b rxa u + 2 b r Xa f ^Du + 2 f b rxa ^Du 24

26 = ^d 2 fu ^d^dfu + e a ^df b rxa u + 2 ^df ^Du + 2 f ^D 2 u = ^r2 fu+ ^d 2 fu+ 2 f ^D2 u+2 = ^r 2 f 6 Rf u +2fu + ^d 2 fu br dgradf u 4 ^df ^Du by the shear-free equation (3) and its integrability condition (3a). If the C -charge of u is q, then the charge of f is q and we have and so ^d 2 f = qff ^d 2 fu = 3fu by (5.34). Thus D 0 (f; u) = ^r 2 f 6 Rf u fu ; and so 0 (f; u) satises the massless Dirac equation if f satises the `Debye potential' equation ^r 2 f 6 Rf = 4 f (49) where = u u is an eigenvector of C with eigenvalue. This equation is of the same form as that for the Debye potential used to construct a Maxwell eld from. However, here the C -charge of the scalar potential is half that required in the Maxwell case, and the eigenvalue is also dierent. In the special case in which the `gauge terms' are zero the shear-free spinor satises the twistor equation. In this case the construction of a massless Dirac solution from a scalar eld satisfying the conformally-covariant wave equation is an example of what Penrose has called `spin raising' [3]. As in the Maxwell case, we may form a scalar potential from a shear-free spinor and a massless Dirac solution. In the special case in which the shear-free equation reduces to the twistor equation this corresponds to Penrose's `spin lowering' [3]. Out of the shear-free spinor u and massless Dirac solution we form a scalar f(u; ) from the scalar product (u; ), f(u; ) =(u; ) : (50) (Unfortunately the notation doesn't work well for us here. The brackets on the left-hand side denote that the scalar is constructed from u and, whereas the bracket on the righthand side denotes the symplectic product of the two spinors.) The scalar f(u; ) has the same C -charge as u, and we may evaluate the gauged Laplacian of it by noting that the gauged covariant derivative is compatible with the spinor product: ^r 2 f(u; )=(^r2 u; )+2(b rx au; r Xa )+(u; r 2 ) : 25

27 Since u satises the shear-free equation (3) ( ^rx au; r Xa )= 4 (ea^du; rxa ) = 4 ( ^Du; D ) =0 since satises the massless Dirac equation. We may then use (23) to relate the spinor Laplacian to the square of the Dirac operator, which gives zero when acting on, togive (^rx au; r Xa )=(^r2 u+ 4 Ru; ) =(u + 6 Ru; ) by the integrability condition (3b), =(+ 6 R)f(u; ) : Thus the scalar f(u; ) satises the `Debye potential' equation (49). Note, however, that f(u; ) has the same `charge' as u, opposite to that required to combine with u to make a massless Dirac solution. To construct one Dirac solution from another we need a pair of shear-free spinors with opposite `charges', and this is just the case in which we have a CKY tensor. In that case we may proceed as in the Maxwell case and dene the symmetry operator L u u 2 by L u u 2 = 0 ((u ; );u 2 ) = ^d(u ; )u (u ; )^Du 2 : (5) By interchanging the two spinors in this construction we could have formed the operator L u2 u. However, as we shall see, these two operators are in fact the same when acting on massless Dirac elds. Since b r is compatible with the spinor inner product, ^d(u ; )=(b rxa u ; )e a +(u ;r Xa )e a Thus = 4 (e a^du ; )e a +(u ;r Xa )e a by(3). ^d(u ; )u 2 = 4 (ea u 2 e a^du )( )+(e a u 2 u )(r Xa ) = 4 ea (u 2 ^Du )e a + 2 ea (u 2 u + u u 2 )(r Xa ) + 2 ea (u 2 u u u 2 )(r Xa ) : (52) 26